Constructions, loci, bearings, plans and elevations – Year 11 GCSE maths worksheet with answers to download as PDF

Bearings are measured clockwise from north and written with three figures.

1. Write down the bearing for each compass direction: east, south, west, south-east. [2 marks]
2. The bearing of \(B\) from \(A\) is \(065^\circ\). Work out the bearing of \(A\) from \(B\). [2 marks]
3. Work out the back bearing of each: \(230^\circ\), \(312^\circ\), \(148^\circ\). [3 marks]
4. Explain the rule for finding a back bearing. [1 mark]

A map has a scale of \(1:25\,000\).

1. Work out the real distance, in kilometres, represented by 4 cm on the map. [2 marks]
2. The real distance between two villages is 3.5 km. What is it on the map, in centimetres? [2 marks]
3. On another map with scale \(1:50\,000\) a lake is 12 cm long. How many kilometres is that? [2 marks]
4. Write the scale \(1:25\,000\) as "1 cm represents ... m". [1 mark]

Use the standard constructions to answer these questions. You should describe the steps.

1. Describe how to construct the perpendicular bisector of a line \(AB\) of length 8 cm. [3 marks]
2. Every point on this bisector is the same distance from \(A\) and \(B\). A point \(P\) is on the bisector and 3 cm from the line \(AB\). How far is \(P\) from \(A\)? [3 marks]
3. Describe how to bisect an angle of \(70^\circ\). What size is each half? [3 marks]

Describe the locus (set of points) in each case.

1. The points that are 3 cm from a fixed point \(P\). [1 mark]
2. The points that are the same distance from two points \(A\) and \(B\). [1 mark]
3. The points that are the same distance from two crossing straight lines. [1 mark]
4. The points that are exactly 2 cm from a straight line segment \(AB\) of length 6 cm. Work out the perimeter of the boundary of the points that are at most 2 cm from \(AB\), to 1 decimal place. [4 marks]

A square-based pyramid has a base of side 6 cm and a vertical height of 4 cm.

1. Describe the plan view (from above). [2 marks]
2. Describe the front elevation. [2 marks]
3. Work out the slant height of one triangular face. [3 marks]
4. Work out the area of one triangular face and then the total area of the four faces. [3 marks]

A ship sails 12 km on a bearing of \(040^\circ\) and then 9 km on a bearing of \(130^\circ\).

1. Show that the two legs are at right angles. [2 marks]
2. Work out how far the ship is from its starting point. [2 marks]
3. Work out the bearing of the ship from the start, to the nearest degree. [3 marks]
4. Work out the bearing of the start from the ship. [1 mark]

A ferry leaves port \(P\), sails 20 km on a bearing of \(060^\circ\) to island \(A\), and then 20 km on a bearing of \(150^\circ\) to island \(B\).

1. Show that angle \(PAB=90^\circ\). [2 marks]
2. Work out the distance \(PB\) to 2 decimal places. [2 marks]
3. Work out the bearing of \(B\) from \(P\). [3 marks]
4. Work out the bearing of \(P\) from \(B\). [1 mark]

A lifeboat station \(H\) is 15 km from a lifeboat \(L\) on a bearing of \(120^\circ\) from \(H\). A ship \(S\) is 8 km from \(L\) on a bearing of \(210^\circ\) from \(L\).

1. Work out the bearing of \(H\) from \(L\). [1 mark]
2. Show that angle \(HLS=90^\circ\). [2 marks]
3. How far is the ship from the station? [2 marks]
4. Work out the bearing of the ship from the station, to 1 decimal place. [3 marks]

A walking map has a scale of \(1:50\,000\). A route has three straight sections measuring 4.2 cm, 6.8 cm and 3.0 cm on the map. The straight-line distance from start to finish measures 9.6 cm.

1. Work out the real length of each section. [3 marks]
2. Work out the total length of the route in kilometres. [2 marks]
3. Walkers go at 4 km/h. How long does the route take? [2 marks]
4. How much shorter is the straight-line distance? [2 marks]

Two trees \(A\) and \(B\) are 10 m apart. Treasure is buried so that it is nearer to \(A\) than to \(B\) and also within 6 m of \(A\). Draw a scale diagram using 1 cm for 1 m.

1. Describe how you would draw the boundary for "nearer to \(A\) than to \(B\)". [2 marks]
2. Describe the boundary for "within 6 m of \(A\)". [1 mark]
3. A spade is at a point on the line \(AB\), 6 m from \(A\). Could the treasure be there? Explain. [2 marks]
4. The treasure is on the line \(AB\). What is the greatest and least distance it can be from \(A\)? [2 marks]

A router reaches 15 m in every direction. A rectangular office is 20 m by 12 m.

1. The router is placed at the centre of the office. Work out the distance from the router to a corner. [3 marks]
2. Does the router cover the whole office? [1 mark]
3. The router is moved to one corner. Does it cover the opposite corner? Show your working. [3 marks]
4. Work out the area of the circle of coverage, in m² to 1 decimal place. [2 marks]

A small plane flies 200 km north and then 150 km east.

1. Work out the direct distance from its starting point. [2 marks]
2. Work out the bearing of its current position from the start, to the nearest degree. [3 marks]
3. Work out the bearing it must fly on to return directly to the start. [1 mark]
4. The plane flies at 250 km/h. How long does the direct return take? [2 marks]

Two radio masts \(A\) and \(B\) are 10 km apart. The signal from \(A\) reaches 6 km and the signal from \(B\) reaches 7 km.

1. Explain why the two signals overlap. [1 mark]
2. A house is on the line \(AB\), 4 km from \(A\). Does it receive both signals? [2 marks]
3. How long is the stretch of the line \(AB\) that receives both signals? [3 marks]
4. A house on the line \(AB\) is 8 km from \(A\). Which signal(s) does it receive? [2 marks]

A garden plan uses a scale of \(1:100\). The lawn is a rectangle 8.5 m by 6 m.

1. What are the dimensions of the lawn on the plan? [2 marks]
2. Work out the real area of the lawn. [1 mark]
3. Turf costs £4.20 per square metre. Work out the cost of turfing the lawn. [2 marks]
4. A fence goes all the way round the lawn. Fencing costs £18 per metre. Work out the cost. [3 marks]

Town \(B\) is 24 km from town \(A\) on a bearing of \(050^\circ\). Town \(C\) is 18 km from \(B\) on a bearing of \(140^\circ\).

1. Show that angle \(ABC=90^\circ\). [3 marks]
2. Work out the distance \(AC\). [2 marks]
3. Work out the bearing of \(C\) from \(A\) to the nearest degree. [3 marks]
4. Work out the bearing of \(A\) from \(C\) to the nearest degree. [2 marks]

\(ABCD\) is a rectangle with \(AB=10\) cm and \(AD=6\) cm. Use axes with \(A\) at \((0,0)\), \(B\) at \((10,0)\) and \(D\) at \((0,6)\). The region \(R\) contains all the points inside the rectangle that are nearer to \(A\) than to \(B\) and are less than 4 cm from \(D\).

1. Describe the boundary line for "nearer to \(A\) than to \(B\)". [2 marks]
2. Describe the other boundary. [1 mark]
3. Is the point \((3,5)\) in \(R\)? Show your working. [3 marks]
4. Is the point \((4,1)\) in \(R\)? Is \((6,5)\) in \(R\)? Explain. [4 marks]

The plan of a prism is an L-shape made from four squares each of side 3 cm (three in a row with one more on top of one end). The prism is 5 cm tall.

1. Sketch the plan and the front elevation (from the long side). [3 marks]
2. Work out the volume of the prism. [3 marks]
3. Work out the total surface area. [4 marks]

Construct triangle \(ABC\) with \(AB=7\) cm, \(AC=6\) cm and angle \(BAC=50^\circ\), then measure \(BC\).

1. Describe the steps of the construction. [3 marks]
2. Use the cosine rule to work out \(BC\) to 2 decimal places. [3 marks]
3. Measuring gives 5.6 cm. Is this consistent with your answer? [1 mark]
4. Bisect angle \(BAC\) and say what angle the bisector makes with \(AB\). [2 marks]

A goat is tied by a 5 m rope to the corner of a rectangular shed that measures 8 m by 6 m. The goat can graze on grass outside the shed. The rope is fixed to the outside corner.

1. Explain why the goat can graze three quarters of a circle. [2 marks]
2. Work out the grazing area to 1 decimal place. [3 marks]
3. Why can the goat not graze around the long side? [2 marks]

The goat is now tied with a 10 m rope to the same corner of the shed (8 m by 6 m).

1. Explain why part of the grazing area is three quarters of a circle of radius 10 m. [1 mark]
2. The goat can go round the 8 m side and the 6 m side. Show that the extra areas are quarter circles of radius 2 m and 4 m. [4 marks]
3. Work out the total grazing area in terms of \(\pi\) and then to 1 decimal place. [5 marks]